# Lattices in $\mathbb C$ as modules of the ring of integers in an imaginary quadratic field

Let $K$ be an imaginary quadratic number field and let $O_K\subset K$ be the ring of algebraic integers in $K$. Let us call a lattice $\Lambda\subset\mathbb C$ normalized if the tori $\mathbb C/\Lambda$ and $\mathbb C/O_K$ have the same area.

The following is true:

A normalized lattice $\Lambda\subset\mathbb C$ is a $O_K$-module (i.e. $\alpha\lambda\in\Lambda$ for all $\alpha\in O_K$, $\lambda\in\Lambda$) if and only if $|\lambda|^2\in\mathbb Z$ for every $\lambda\in\Lambda$.

What is a proof of this statement (if possible direct/elegant/short)? Are there any similar/analogous results?

It is easy to see the $\Rightarrow$ implication: if a normalized lattice $\Lambda\subset\mathbb C$ is a $O_K$ module then for every $\lambda\in\Lambda$ we see that $\lambda O_K\subset\Lambda$, and since $\Lambda$ is normalized, we get (if $\lambda\neq0$) $|\lambda|^2=[\Lambda:\lambda O_K]$, hence $|\lambda|^2\in\mathbb Z$. What is, however, a proof of the $\Leftarrow$ implication? (I can prove it with a random-looking calculation, but a conceptual proof should exist.)

context: The statement is equivalent to the description of the ideal class group of $K$ in terms of quadratic forms (any $\Lambda$ gives us a quadratic form, namely $\lambda\mapsto|\lambda|^2$). I just want to understand this description and its proof geometrically (so everything should be well-known, just not to me).

Your $\Lambda$ carries an integral-valued quadratic form, and I would start by proving that$$\mathbb{Q} \Lambda := \{\text{rational multiples of }\Lambda\}$$is a $1$-dimensional $K$-vector space inside the field of complex numbers; this is done by considering the determinant of the inner product on $\Lambda$. Next I would consider the ring $R$ (say) of all $x$ in $K$ with $x\Lambda \subset \Lambda$. That is a subring of your $\mathcal{O}_K$ of finite index. Now there is a subtlety, which derives from an algebraic property of $R$ (it is a Gorenstein ring) but that can also be proved by an unpleasant explicit calculation (I saw it once in Borevich and Shafarevich's book on number theory), which says that $\Lambda$ is "locally" generated by one element over $R$. In explicit terms, this means that for every positive integer $n$ there exists $\lambda$ in $\Lambda$ for which $[\Lambda : R\lambda]$ is coprime to $n$. Now, by your own calculation,$$|\lambda|^2 = [\Lambda : R\lambda]\, {{\text{covol}(\Lambda)}\over{\text{covol}(R)}} = {{[\Lambda : R\lambda]}\over{[\mathcal{O}_K: R]}}.$$Choosing $n = [\mathcal{O}_K : R]$ and using that $|\lambda|^2$ is an integer, one deduces that $\mathcal{O}_K = R$, as required.
The Gorenstein property derives from $R$ being monogenic over the ring $\mathbb{Z}$ of integers. It is a specific property for quadratic rings, and makes it hard to generalize the proof and probably even the statement to higher dimensional situations.